User:ECCclass/Justesen code

In coding theory, Justesen codes form a class of error-correcting codes which are derived from Reed-Solomon codes and have good error-control properties.

Definition
Let R be a Reed-Solomon code of length N = 2m &minus; 1, rank K and minimum weight N &minus; K + 1. The symbols of R are elements of F = GF(2m) and the codewords are obtained by taking every polynomial &fnof; over F of degree less than K and listing the values of &fnof; on the non-zero elements of F in some predetermined order. Let α be a primitive element of F. For a codeword a = (a1, ..., aN) from R, let b be the vector of length 2N over F given by


 * $$ \mathbf{b} = \left( a_1, a_1, a_2, \alpha^1 a_2, \ldots, a_N, \alpha^{N-1} a_N \right) $$

and let c be the vector of length 2N m obtained from b by expressing each element of F as a binary vector of length m. The Justesen code is the linear code containing all such c.

Properties
The parameters of this code are length 2m N, dimension m K and minimum distance at least


 * $$ \sum_{i=1}^\ell i \binom{2m}{i} . $$

The Justesen codes are examples of concatenated codes.